Planning of life-depleting preventive maintenance activities with replacements

We consider a system that generates a reward at some predefined decreasing rate as its virtual age increases. Periodic maintenance (PM) is performed to restore the system to as-good-as new condition, i.e., its virtual age is decreased to zero after each maintenance action. However, the system has a given deterministic initial lifetime and each maintenance action also results in a depletion of the remaining system’s lifetime. Whenever the lifetime expires, the system is replaced by a new identical one at some predefined cost. From the application perspective the considered setting is primarily motivated by the practice of battery-powered systems, e.g., sensors, that need to be remotely monitored. We formulate a general repair-replacement problem with an infinite time horizon, where we seek an optimal number of PM actions and an optimal PM interval. We describe global optimization techniques for solving a general case of the problem under some reasonably mild assumptions on the reward and lifetime depletion rates. For the special case of a constant lifetime depletion per each maintenance action, some analytic observations are provided. Finally, we also demonstrate that for sufficiently large replacement costs the infinite-horizon problem can be solved by leveraging the results for the related finite-horizon problem previously considered in the literature.